Hartshorne, Algebraic Geometry

SRC-TEXT-HART77

Area: Algebraic Geometry
Status: draft
Source

Contents

442 problems.

I.1

Coordinate rings of the parabola, the hyperbola, and any conic — I.1.1
The twisted cubic is an affine variety of dimension one — I.1.2
Decomposing \(V(x^2-yz,\ xz-x)\) into irreducible components — I.1.3
The Zariski topology on \(\AA^2\) against the product topology — I.1.4
Which \(k\)-algebras are affine coordinate rings — I.1.5
Opens are dense in an irreducible space, and closures stay irreducible — I.1.6
Four characterisations of a Noetherian space, and what they force — I.1.7
A hypersurface section drops the dimension by exactly one — I.1.8
\(r\) generators bound the codimension of every component by \(r\) — I.1.9
Dimension under subspaces, open covers, and closed subsets — I.1.10
The curve \((t^3, t^4, t^5)\) is not a local complete intersection — I.1.11*
An irreducible real polynomial with reducible zero set — I.1.12

I.2

The homogeneous Nullstellensatz — I.2.1
The irrelevant ideal and the empty projective vanishing locus — I.2.2
Basic properties of the operators \(Z\) and \(I\) in projective space — I.2.3
The inclusion-reversing correspondence for projective algebraic sets — I.2.4
\(\PP^n\) is noetherian, and the irreducible components of a projective algebraic set — I.2.5
The homogeneous coordinate ring satisfies \(\dim S(Y) = \dim Y + 1\) — I.2.6
\(\dim \PP^n = n\), and a quasi-projective variety has the dimension of its closure — I.2.7
A projective variety has codimension one exactly when it is a hypersurface — I.2.8
Projective closure of an affine variety, and the twisted cubic curve — I.2.9
The affine cone over a projective variety — I.2.10
Linear varieties in \(\PP^n\) and the dimension of their intersections — I.2.11
The \(d\)-uple embedding of \(\PP^n\) in \(\PP^N\) — I.2.12
Every curve on the Veronese surface is cut out by a hypersurface — I.2.13
The Segre embedding of \(\PP^r \times \PP^s\) — I.2.14
The quadric surface \(xy = zw\) in \(\PP^3\) and its two rulings — I.2.15
Intersections of varieties, and why \(I(Y \intersect Z) \neq I(Y) + I(Z)\) — I.2.16
Strict versus set-theoretic complete intersections — I.2.17

I.3

Conics, and varieties that are not isomorphic to one another — I.3.1
A bijective bicontinuous morphism need not be an isomorphism — I.3.2
Morphisms and the induced maps on local rings — I.3.3
The \(d\)-uple embedding is an isomorphism onto its image — I.3.4
The complement of a hypersurface in \(\PP^n\) is affine — I.3.5
The punctured plane \(\AA^2 \sm \ts{(0,0)}\) is quasi-affine but not affine — I.3.6
A positive-dimensional projective variety meets every hypersurface — I.3.7
Regular functions off a codimension two linear subspace of \(\PP^n\) are constant — I.3.8
The homogeneous coordinate ring is not an isomorphism invariant — I.3.9
Subvarieties and the restriction of a morphism — I.3.10
Primes of \(\mco_P\) correspond to closed subvarieties through \(P\) — I.3.11
The local ring at a point satisfies \(\dim \mco_P = \dim X\) — I.3.12
The local ring of a subvariety — I.3.13
Projection from a point, and the cuspidal cubic image of the twisted cubic — I.3.14
Products of affine varieties and the tensor product of coordinate rings — I.3.15
Products of quasi-projective varieties via the Segre embedding — I.3.16
Normal varieties and the normalization of an affine variety — I.3.17
Projective normality depends on the embedding, via the twisted quartic — I.3.18
An automorphism of \(\AA^n\) has constant Jacobian — I.3.19
Extending a regular function across a normal point in dimension at least two — I.3.20
Group varieties \(\GG_a\) and \(\GG_m\), and the group structure on \(\Hom(X,G)\) — I.3.21

I.4

Gluing regular functions and the domain of definition of a rational function — I.4.1
Largest open set on which a rational map is a morphism — I.4.2
Domain of \(f = x_1/x_0\) on \(\PP^2\) and of the induced map to \(\PP^1\) — I.4.3
Conics, the cuspidal cubic, and the nodal cubic are rational curves — I.4.4
The quadric surface \(xy = zw\) is birational but not isomorphic to \(\PP^2\) — I.4.5
The standard quadratic plane Cremona transformation — I.4.6
Isomorphic local rings at points give isomorphic open neighborhoods — I.4.7
A variety of positive dimension has the cardinality of \(k\) — I.4.8
Projection from a general point is birational onto its image — I.4.9
Blowing up the cusp of \(y^2 = x^3\) gives a bijective non-isomorphism — I.4.10

I.5

Singular points of four plane quartics — I.5.1
Singular points of three affine surfaces in \(\AA^3\) — I.5.2
Multiplicity \(\mu_P(Y)\) of a point on a plane curve and its tangent directions — I.5.3
Intersection multiplicity \((Y \cdot Z)_P\) and Bézout for a line — I.5.4
A nonsingular plane curve of each degree in each characteristic — I.5.5
Resolving nodes, tacnodes, and the cusp \(y^3 = x^5\) by blowing up — I.5.6
Blowing up the vertex of the cone over a nonsingular plane curve — I.5.7
Jacobian rank criterion for nonsingularity in \(\PP^n\) — I.5.8
Nonvanishing partials force \(f\) to be irreducible — I.5.9
The Zariski tangent space \(T_P(X) = \dualof{(\mfm/\mfm^2)}\) — I.5.10
The elliptic quartic curve in \(\PP^3\) projects to a plane cubic — I.5.11
Rank normal form, irreducibility, and singular locus of a quadric hypersurface — I.5.12
The nonnormal locus of a variety is a proper closed subset — I.5.13
Analytically isomorphic plane curve singularities and formal factorization — I.5.14
The space \(\PP^N\) of plane curves of degree \(d\) and its nonsingular locus — I.5.15

I.6

I.7

II.1

The constant sheaf is the sheafification of the constant presheaf — II.1.1
Kernels, images, and exactness of sheaves are computed on stalks — II.1.2
Surjectivity of a sheaf morphism is a local lifting condition — II.1.3
Sheafification preserves injectivity, and the image is a subsheaf — II.1.4
A morphism of sheaves is an isomorphism exactly when it is injective and surjective — II.1.5
A subsheaf gives a short exact sequence with its quotient sheaf — II.1.6
The image and cokernel of a sheaf morphism as quotient sheaves — II.1.7
The global sections functor is left exact — II.1.8
The direct sum of two sheaves is both a product and a coproduct — II.1.9
Direct limits of sheaves are the sheafification of the sectionwise limit — II.1.10
On a noetherian space the sectionwise direct limit of sheaves is already a sheaf — II.1.11
Inverse limits of sheaves are computed section by section — II.1.12
Sheafification as the sheaf of continuous sections of the espace étalé — II.1.13
The support of a section is closed, but the support of a sheaf need not be — II.1.14
Local morphisms between two sheaves form a sheaf — II.1.15
Flasque sheaves and the exactness of sections — II.1.16
Skyscraper sheaves as pushforwards from the closure of a point — II.1.17
Inverse image is left adjoint to direct image — II.1.18
Extending a sheaf by zero outside a closed or an open subset — II.1.19
The subsheaf of sections supported in a closed subset — II.1.20
Ideal sheaves on varieties and the failure of exactness of global sections — II.1.21
Glueing sheaves along an open cover with cocycle data — II.1.22

II.2

Distinguished open subsets of an affine scheme are affine — II.2.1
Open subsets of a scheme carry an induced scheme structure — II.2.2
Reducedness is local, and every scheme has a reduction — II.2.3
Morphisms to an affine scheme are ring maps into global sections — II.2.4
The spectrum of the integers is the terminal scheme — II.2.5
The spectrum of the zero ring is the initial scheme — II.2.6
Morphisms from the spectrum of a field are points with residue field inclusions — II.2.7
Dual numbers detect Zariski tangent vectors — II.2.8
Irreducible closed subsets of a scheme have unique generic points — II.2.9
The spectrum of the polynomial ring over the reals — II.2.10
The spectrum of a polynomial ring over a finite field — II.2.11
Glueing a family of schemes along open subschemes — II.2.12
Quasi-compactness of spectra and noetherian topological spaces — II.2.13
Graded ring homomorphisms induce morphisms of Proj — II.2.14
Varieties over an algebraically closed field embed fully faithfully into schemes — II.2.15
Sections over the nonvanishing locus of a global function — II.2.16
A criterion for affineness via distinguished opens generating the unit ideal — II.2.17
Injectivity and surjectivity of a ring map read off from the induced morphism — II.2.18
A spectrum is disconnected exactly when the ring splits as a product — II.2.19

II.3

Locally of finite type can be checked on every open affine of the target — II.3.1
Quasi-compactness of a morphism can be checked on every open affine — II.3.2
Finite type equals locally of finite type together with quasi-compact — II.3.3
Finiteness of a morphism can be checked on every open affine of the target — II.3.4
Finite morphisms are quasi-finite and closed, but not conversely — II.3.5
The local ring at the generic point of an integral scheme is its function field — II.3.6
A dominant generically finite morphism becomes finite over a dense open set — II.3.7
Normalization of an integral scheme and its universal property — II.3.8
The underlying space of a product of schemes is not the product of the spaces — II.3.9
Fibres of a morphism as schemes over the residue field — II.3.10
Closed subschemes under base change and the scheme-theoretic image — II.3.11
Closed subschemes of Proj from surjections of graded rings — II.3.12
Permanence properties of morphisms of finite type — II.3.13
Closed points are dense in a scheme of finite type over a field — II.3.14
Geometric irreducibility, reducedness, and integrality after base change — II.3.15
Noetherian induction on closed subsets — II.3.16
Zariski spaces, closed points, and stability under specialization — II.3.17
Constructible subsets of a Zariski space — II.3.18
Images of constructible sets under a finite type morphism are constructible — II.3.19
Dimension of an integral scheme of finite type over a field — II.3.20
Dimension theory fails for the affine line over a discrete valuation ring — II.3.21
Dimension of the fibres of a dominant morphism — II.3.22*
The product of two varieties agrees with the fibre product of the associated schemes — II.3.23

II.4

Finite morphisms are proper — II.4.1
Morphisms into a separated scheme agreeing on a dense open subset — II.4.2
Intersections of open affines in a separated scheme — II.4.3
The image of a proper closed subscheme is proper — II.4.4
Centers of valuations on separated and proper schemes — II.4.5
Proper morphisms of affine varieties are finite — II.4.6
Real forms of complex schemes and semilinear involutions — II.4.7
Stability properties of a class of morphisms — II.4.8
Compositions of projective morphisms are projective — II.4.9
Chow’s lemma — II.4.10
Valuative criteria using discrete valuation rings — II.4.11
Valuation rings of function fields of dimension one and two — II.4.12

II.5

Duals of locally free sheaves and the projection formula — II.5.1
Modules over the spectrum of a discrete valuation ring — II.5.2
The tilde and global sections functors are adjoint — II.5.3
Quasi-coherence as a local cokernel of free sheaves — II.5.4
Pushforward of coherent sheaves along a finite morphism — II.5.5
Support of a module and sections with support in a closed set — II.5.6
Local freeness from free stalks, and invertible sheaves — II.5.7
Semicontinuity of the fibre dimension of a coherent sheaf — II.5.8
The comparison map from a graded module to its sections — II.5.9
Saturated ideals and closed subschemes of projective space — II.5.10
The Cartesian product of graded rings and the Segre embedding — II.5.11
Tensor products of very ample invertible sheaves — II.5.12
The Veronese subring and the \(d\)-uple embedding — II.5.13
Projective normality of the \(d\)-uple embedding — II.5.14
Extension of coherent sheaves from an open subset — II.5.15
Tensor, symmetric, and exterior powers of locally free sheaves — II.5.16
Affine morphisms and the relative spectrum — II.5.17
Vector bundles and locally free sheaves — II.5.18

II.6

The class group of a product with projective space — II.6.1
Degrees of divisors on subvarieties of projective space — II.6.2
The class group of an affine cone — II.6.3
Integral closedness of a square-root extension — II.6.4
Class groups of quadric hypersurfaces and Klein’s theorem — II.6.5
The group law on a nonsingular plane cubic — II.6.6
Degree-zero Cartier classes on a nodal cubic — II.6.7
Pullback of invertible sheaves on Picard groups — II.6.8
The Picard group of a singular curve via normalization — II.6.9
The Grothendieck group of a noetherian scheme — II.6.10
The Grothendieck group of a nonsingular curve — II.6.11
Degree of a coherent sheaf on a complete nonsingular curve — II.6.12

II.7

A surjection of invertible sheaves is an isomorphism — II.7.1
Morphisms from the same linear system differ by a linear projection — II.7.2
Morphisms between projective spaces — II.7.3
Schemes with an ample invertible sheaf are separated — II.7.4
Properties of ample and very ample invertible sheaves — II.7.5
The Riemann-Roch problem for multiples of a divisor — II.7.6
The Veronese surface and rational surfaces from linear systems — II.7.7
Sections of a projective bundle and invertible quotients — II.7.8
The Picard group of a projective bundle — II.7.9
Projective n-space bundles over a scheme — II.7.10
Ideal sheaves giving isomorphic blowings-up — II.7.11
Strict transforms after blowing up an intersection — II.7.12
A complete nonprojective variety — II.7.13
Very ampleness on a relative Proj — II.7.14

II.8

Differentials at nonclosed points and regularity — II.8.1
A nowhere-vanishing section of a high-rank locally free sheaf — II.8.2
Differentials and canonical sheaves of product schemes — II.8.3
Complete intersections in projective space and their genus — II.8.4
Picard group and canonical sheaf of a blowing-up — II.8.5
The infinitesimal lifting property — II.8.6
Infinitesimal extensions of a scheme by a coherent sheaf — II.8.7
Plurigenera and regular differential forms are birational invariants — II.8.8

II.9

Formal-regular functions along a subvariety of projective space — II.9.1
A morphism contracting a subvariety contracts the whole space — II.9.2
Global sections on an affine formal scheme are exact — II.9.3
Coherence of the middle term of an exact sequence — II.9.4
Finitely many global generators for a coherent sheaf — II.9.5
The Picard group of a noetherian formal scheme — II.9.6

III.2

Nonvanishing top cohomology of extensions by zero on affine space — III.2.1, part (b)*
Flasque resolution of the structure sheaf on the projective line — III.2.2
Cohomology with supports in a closed subset — III.2.3
Mayer-Vietoris sequence for cohomology with supports — III.2.4
Local cohomology at a point is computed on the local space — III.2.5
Direct limits of injective sheaves on a noetherian space are injective — III.2.6
Derived functor cohomology of the circle — III.2.7

III.3

Affineness is detected by the reduced scheme — III.3.1
Affineness is detected on irreducible components — III.3.2
Local cohomology modules of an ideal — III.3.3
Cohomological interpretation of depth — III.3.4
Depth two and unique extension of sections across a point — III.3.5
Injective quasi-coherent sheaves and cohomology — III.3.6, part (b)*
Deligne’s formula for sections on open subsets of an affine scheme — III.3.7
An injective module whose localization map is not surjective — III.3.8

III.4

Cohomology is preserved by pushforward along an affine morphism — III.4.1
Chevalley’s theorem on finite surjective images of affine schemes — III.4.2
Cohomology of the punctured affine plane — III.4.3
Čech cohomology in the limit over coverings computes \(H^1\) — III.4.4
The Picard group as first cohomology of the sheaf of units — III.4.5
Picard groups under a square-zero thickening — III.4.6
Čech computation of the cohomology of a plane curve — III.4.7
Cohomological dimension of a noetherian separated scheme — III.4.8, part (d)*
Two planes meeting at a point are not a set-theoretic complete intersection — III.4.9
Infinitesimal extensions classified by \(H^1\) of the twisted tangent sheaf — III.4.10*
Čech cohomology agrees with derived functor cohomology on acyclic covers — III.4.11

III.5

Additivity of the Euler characteristic — III.5.1
Existence of the Hilbert polynomial of a coherent sheaf — III.5.2
Arithmetic genus and nonsingular projective curves — III.5.3
Hilbert polynomial homomorphism on the Grothendieck group — III.5.4
Cohomology of a complete intersection in projective space — III.5.5
Curves on a nonsingular quadric surface — III.5.6
Ampleness under restriction, reduction, and finite surjections — III.5.7
Every one-dimensional proper scheme is projective — III.5.8
A nonprojective infinitesimal extension of the projective plane — III.5.9
Exactness of global sections of a complex after a large twist — III.5.10

III.6

Extensions of sheaves of modules are classified by \(\Ext^1\) — III.6.1
No projective object surjects onto the structure sheaf of the projective line — III.6.2
Coherence and quasi-coherence of sheaf Ext — III.6.3
Sheaf Ext as a universal delta-functor — III.6.4
Homological dimension of a coherent sheaf — III.6.5
Projective dimension over a regular local ring detected by Ext into the ring — III.6.6
Ext on an affine scheme computes module Ext — III.6.7
Kleiman’s theorem on enough locally free sheaves — III.6.8
Grothendieck groups of coherent and locally free sheaves agree — III.6.9
Duality for a finite flat morphism — III.6.10

III.7

Vanishing of global sections of an inverse ample sheaf — III.7.1
Dualizing sheaves under a finite morphism — III.7.2
Hodge cohomology of projective space — III.7.3
The cohomology class of a subvariety — III.7.4*

III.8

Vanishing higher direct images give isomorphic cohomology — III.8.1
Cohomology is preserved by an affine morphism — III.8.2
The projection formula for higher direct images — III.8.3
Cohomology of a projective bundle and geometrically ruled surfaces — III.8.4

III.9

Flat morphisms of finite type are open — III.9.1
An embedded point at the cusp of a plane cubic — III.9.2
Examples of flatness and nonflatness — III.9.3
The flat locus is open — III.9.4
Very flat families and projective cones — III.9.5
Complete intersections detected by a hyperplane section — III.9.6
Infinitesimal deformations of a closed subscheme and the normal sheaf — III.9.7
The module \(T^1\) classifies infinitesimal deformations of an algebra — III.9.8*
Two planes meeting at a point form a rigid algebra — III.9.9
Rigidity of the projective line and its global deformations — III.9.10, part (c)*
Castelnuovo’s genus bound for a space curve of degree \(d\) — III.9.11

III.10

Smooth is not regular over a nonperfect field — III.10.1
Smoothness spreads out from a smooth fibre — III.10.2
Characterizations of an etale morphism — III.10.3
Etale morphisms via completed local rings — III.10.4
Local freeness is detected on etale neighborhoods — III.10.5
An etale double cover of the nodal cubic — III.10.6
Serre’s linear system with moving singularities — III.10.7
A pencil of cones with a moving singularity in the base locus — III.10.8
Miracle flatness for a Cohen-Macaulay source — III.10.9

III.11

Higher direct images on punctured affine space — III.11.1
A projective quasi-finite morphism is finite — III.11.2
Connectedness of divisors in a base point free linear system — III.11.3
The principle of connectedness for a flat family — III.11.4
The Picard group of a formal completion along a hypersurface — III.11.5*
Algebraizability of locally free sheaves on a formal completion — III.11.6
A nonalgebraizable invertible sheaf on a formal completion of the plane — III.11.7
Vanishing on one fibre forces a higher direct image to vanish nearby — III.11.8

III.12

Embedding dimension is upper semicontinuous — III.12.1
Cohomology of a family of hypersurfaces of fixed degree is constant — III.12.2
Jumping cohomology for the ideal sheaf of a degenerating quartic curve — III.12.3
Fibrewise isomorphic invertible sheaves differ by a pullback — III.12.4
The Picard group of a projective bundle — III.12.5
The Picard group of a product with vanishing first cohomology — III.12.6*

IV.1

Existence of a rational function regular away from a single point — IV.1.1
A rational function with poles at a prescribed finite set of points — IV.1.2
A regular one-dimensional scheme that is not proper over \(k\) is affine — IV.1.3
A one-dimensional scheme with no proper irreducible component is affine — IV.1.4
\(\dim \abs{D} \leq \deg D\) for an effective divisor, with equality iff \(D=0\) or \(g=0\) — IV.1.5
Every curve of genus \(g\) admits a finite map to \(\PP^1\) of degree \(\leq g+1\) — IV.1.6
Hyperelliptic curves exist in every genus \(g \geq 2\) — IV.1.7
The arithmetic genus of a singular curve and the local invariants \(\delta_P\) — IV.1.8, part (c)*
Riemann-Roch for singular curves — IV.1.9*
\(\Pic X\) is in bijection with \(X_{\reg}\) when \(p_a(X)=1\) — IV.1.10

IV.2

IV.3

On a curve of genus \(2\), a divisor is very ample iff \(\deg D \geq 5\) — IV.3.1
A plane quartic is not hyperelliptic and its canonical divisors are the line sections — IV.3.2
The canonical divisor of a complete intersection curve is very ample — IV.3.3
The rational normal curve of degree \(d\) in \(\PP^d\) — IV.3.4
A space curve projects to a singular plane curve, and \(g<\frac{1}{2}(d-1)(d-2)\) — IV.3.5
Classification of curves of degree \(4\) in projective space — IV.3.6
A nodal plane curve need not be a projection of a nonsingular space curve — IV.3.7
Strange curves exist only in positive characteristic — IV.3.8
Bertini’s lemma on general plane sections of a space curve — IV.3.9
A general \((n-2)\)-plane spanned by points of \(X\) meets \(X\) in no further point — IV.3.10
Projection embeds a smooth \(r\)-fold into \(\PP^{n-1}\) when \(n>2r+1\) — IV.3.11
Existence of plane curves of degree \(d\) with exactly \(r\) nodes — IV.3.12

IV.4

The section ring \(\bigoplus_n H^0(\OO_X(nP))\) of an elliptic curve is a Weierstrass ring — IV.4.1
An elliptic curve embedded by \(\abs{D}\) with \(\deg D \geq 3\) is projectively normal — IV.4.2
Automorphisms of \((X,P_0)\) come from affine substitutions \(x'=ax+b\), \(y'=cy\) — IV.4.3
The \(j\)-invariant is a rational function of the Weierstrass coefficients over \(\QQ\) — IV.4.4
Exactly three values of \(j\) admit an endomorphism of degree \(2\) — IV.4.5
Counting inflection and hyperosculation points, and the \(d^2\) points of order \(d\) — IV.4.6
The dual isogeny \(\hat f\) satisfies \(f \circ \hat f = n\) and \(\deg \hat f = \deg f\) — IV.4.7, part (d)*
The algebraic fundamental group of an elliptic curve — IV.4.8
Isogeny is an equivalence relation with countable isogeny classes — IV.4.9
\(\Pic(X \times X)\) is an extension of \(\operatorname{End}(X, P_0)\) by the pullbacks of \(\Pic X\) — IV.4.10
Complex multiplication by \(\alpha\) has degree \(\abs{\alpha}^2\) and dual \(\bar\alpha\) — IV.4.11
The values of \(\tau\) giving extra automorphisms or a degree \(2\) endomorphism — IV.4.12
The unique \(j\) with vanishing Hasse invariant when \(p=13\) — IV.4.13
The primes where the Fermat cubic has Hasse invariant \(0\) have density \(\frac{1}{2}\) — IV.4.14
The \(p\)-torsion subgroup is \(\ZZ/p\) exactly when the Hasse invariant is \(1\) — IV.4.15
Hasse’s bound \(\abs{a} \leq 2\sqrt{q}\) for \(N = \size X(\FF_q)\) — IV.4.16
Computing \(nP\) on the curve \(y^2+y=x^3-x\) — IV.4.17
The \(26\) integral points of \(y^2=x^3-7x+10\) and their generators — IV.4.18
Reduction mod \(p\) injects the prime-to-\(p\) torsion of \(X(\QQ)\) — IV.4.19
The endomorphism ring in characteristic \(p\) and the field of definition of \(j\) — IV.4.20
Every order in an imaginary quadratic field has the form \(\ZZ + f \cdot \OO\) — IV.4.21
A family of elliptic curves over \(\AA^1_\CC\) with a section is trivial — IV.4.22*

IV.5

A hyperelliptic curve is never a complete intersection — IV.5.1
\(\Aut X\) is finite for a curve of genus \(\geq 2\) in characteristic \(0\) — IV.5.2
Dimensions of the families of curves of genus \(4\) — IV.5.3
Least degree of a nodal plane model of a nonhyperelliptic curve of genus \(4\) — IV.5.4
Canonical models of curves of genus \(5\) and the trisecant cubic surface — IV.5.5
A nonsingular plane quintic has no \(g_3^1\) — IV.5.6
Automorphisms of genus \(3\) curves and the Klein quartic with \(168\) automorphisms — IV.5.7

IV.6

V.1

V.2

Uniqueness of the base curve of a birationally ruled surface — V.2.1
Decomposability of a rank two bundle via disjoint sections — V.2.2
Locally free sheaves on a curve are successive extensions of invertible sheaves — V.2.3
Possible self-intersection numbers of sections of \(C \times \PP^1\) — V.2.4
Which values of the invariant \(e\) occur for ruled surfaces over a curve — V.2.5
Splitting of locally free sheaves on the projective line — V.2.6
Sections of self-intersection one on the elliptic ruled surface — V.2.7
Stability of rank two bundles and classification of the unstable indecomposables — V.2.8
Degree and genus of nonsingular curves on a quadric cone — V.2.9
Canonical curves on a rational scroll and nonhyperelliptic curves with a \(g^1_3\) — V.2.10
Base points and very ampleness for sections of a ruled surface — V.2.11
Base points and very ampleness on a ruled surface over an elliptic curve — V.2.12
Existence of elliptic scrolls of given degree — V.2.13
Irreducible curves and ample divisors on a ruled surface in characteristic \(p\) — V.2.14
A ruled surface in characteristic 3 with a nonample divisor satisfying the numerical criterion — V.2.15
Grothendieck group classes and isomorphism of bundles on an affine curve — V.2.16
Splitting type of the conormal bundle of rational space curves — V.2.17*

V.3

Invariance of the arithmetic genus under blowing up a nonsingular subvariety — V.3.1
Intersection multiplicity drops by the product of multiplicities under a blowup — V.3.2
Ampleness of \(2\pi^*D - E\) on a blowup — V.3.3
The Hilbert-Samuel polynomial and the multiplicity of a local ring — V.3.4
Resolving the point at infinity of a hyperelliptic plane model — V.3.5
Analytic isomorphism of curve singularities versus equivalence — V.3.6
Embedded resolutions and \(\delta_P\) for a list of plane curve singularities — V.3.7
Inequivalent singularities with matching multiplicity data — V.3.8

V.4

Conics through two base points and the quadric surface as a blown up plane — V.4.1
Degree and multiplicities of a strict transform under a quadratic transformation — V.4.2
Reduction of a plane curve to ordinary singularities by quadratic transformations — V.4.3
Associativity of the group law on a plane cubic — V.4.4
Pascal’s theorem for six points on a conic — V.4.5
Three determined base points for quartics through thirteen points — V.4.6
Maximal arithmetic genus of a divisor of given degree on a cubic surface — V.4.7
Divisor classes on the cubic surface containing an irreducible curve — V.4.8*
Minimal positive genus of a curve of given degree on the cubic surface — V.4.9
A numerical inequality underlying the ampleness criterion on the cubic surface — V.4.10
Weyl groups and the automorphisms of the configuration of 27 lines — V.4.11, part (c)*
Kodaira vanishing for the cubic surface — V.4.12
The 16 lines on the degree four Del Pezzo surface and its equations — V.4.13
Existence of space curves of degree at most ten with prescribed genus — V.4.14
Points in general position and exceptional curves on blowups of the plane — V.4.15, part (e)*
The 27 lines and the automorphism group of the Fermat cubic surface — V.4.16

V.5

V.6

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